Möbius inversion formula
The classic
Möbius inversion formula was introduced into
number theory during the 19th century by August Ferdinand Möbius. It was later generalized to other "Möbius inversion formulas"; see
incidence algebra. The classic version states that if
g(
n) and
f(
n) are arithmetic functions satisfying

then

where μ is the
Möbius function and the sums extend over all positive divisors
d of
n.
The formula is also correct if f and g are functions from the positive integers into some abelian group.
In the language of convolutions (see multiplicative function), the inversion formula can also be expressed as
 μ * 1 = ε.
An equivalent formulation of the inversion formula more useful in
combinatorics is as follows: suppose
F(
x) and
G(
x) are
complexvalued
functions defined on the
interval [1,∞) such that

then

Here the sums extend over all positive integers
n which are less than or equal to
x.
The Möbius inversion treated above is the original Möbius inversion. When the partially ordered set of natural numbers ordered by divisibility one is replaced by other locally finite partially ordered sets, one has other Möbius inversion formulas; for an account of those, see incidence algebra.
See also: August Ferdinand Möbius.